Apparent Paradox
When the properties of Gabriel's Horn were discovered, the fact that the rotation of an infinite curve about the x-axis generates an object of finite volume was considered paradoxical. However, the explanation is that the bounding curve, is simply a special case–just like the simple harmonic series (Σx−1)–for which the successive area 'segments' do not decrease rapidly enough to allow for convergence to a limit. For volume segments however, and in fact for any generally constructed higher degree curve (e.g. y = 1/x1.001), the same is not true and the rate of decrease in the associated series is sufficiently rapid for convergence to a (finite) limiting sum.
The apparent paradox formed part of a great dispute over the nature of infinity involving many of the key thinkers of the time including Thomas Hobbes, John Wallis and Galileo Galilei.
Read more about this topic: Gabriel's Horn
Famous quotes containing the words apparent and/or paradox:
“The apparent rulers of the English nation are like the imposing personages of a splendid procession: it is by them the mob are influenced; it is they whom the spectators cheer. The real rulers are secreted in second-rate carriages; no one cares for them or asks after them, but they are obeyed implicitly and unconsciously by reason of the splendour of those who eclipsed and preceded them.”
—Walter Bagehot (1826–1877)
“The paradox of education is precisely this—that as one begins to become conscious one begins to examine the society in which he is being educated.”
—James Baldwin (1924–1987)